On Shishkin meshes, standard numerical methods such as finite element method and finite difference method have been well analyzed [1]}. These methods cannot compute the gradient of the solution accurately. However, in many physical applications, the gradient
is also very important because it is related to some important physical quantities, such as surface friction and temperature and so on. To meet this requirement, the local discontinuous Galerkin (LDG) method [2]} is a good candidate.
The LDG method, as a kind of discontinuous Galerkin (DG) method, inherits many advantages of the DG method [3]}, such as good stability, high-order accuracy and flexibility on \(hp\)  adaptivity. These advantages make the LDG method more attractive to deal with those solutions with layers.
In recent years, the LDG method has been used to singularly perturbed problems; see [4]}, [5]}, [6]}, [7]}.
